Page 1 Chapter 1 Magnetic Materials 1.1 Preliminaries 1.1.1 Required Knowledge â€¢ Magnetism â€¢ Electron spin â€¢ Atom â€¢ Angular momentum (quantum) â€¢ Statistical mechanics 1.1.2 Reading â€¢ Hook and Hall 7.1-7.3, 8.1-8.7 1.2 Introduction â€¢ Magnettechnologyhasmadeenormousadvancesinrecentyearsâ€“without the reductions in size that have come with these advances many modern devices would be impracticable. 1 Page 2 Chapter 1 Magnetic Materials 1.1 Preliminaries 1.1.1 Required Knowledge â€¢ Magnetism â€¢ Electron spin â€¢ Atom â€¢ Angular momentum (quantum) â€¢ Statistical mechanics 1.1.2 Reading â€¢ Hook and Hall 7.1-7.3, 8.1-8.7 1.2 Introduction â€¢ Magnettechnologyhasmadeenormousadvancesinrecentyearsâ€“without the reductions in size that have come with these advances many modern devices would be impracticable. 1 2 CHAPTER 1. MAGNETIC MATERIALS â€¢ The important quantity for many purposes is the energy density of the magnet. 1.3 Magnetic properties - reminder â€¢ There are two ?elds to consider: â€“ The magnetic ?eld H which is generated by currents according to Ampèreâ€™s law. H is measured in A m -1 (Oersteds in old units) â€“ The magnetic induction, or magnetic ?ux density,B, which gives the energy of a dipole in a ?eld, E =-m.B and the torque experienced by a dipole momentm asG=m×B. B is measured in Wb m -2 or T (Gauss in old units). Page 3 Chapter 1 Magnetic Materials 1.1 Preliminaries 1.1.1 Required Knowledge â€¢ Magnetism â€¢ Electron spin â€¢ Atom â€¢ Angular momentum (quantum) â€¢ Statistical mechanics 1.1.2 Reading â€¢ Hook and Hall 7.1-7.3, 8.1-8.7 1.2 Introduction â€¢ Magnettechnologyhasmadeenormousadvancesinrecentyearsâ€“without the reductions in size that have come with these advances many modern devices would be impracticable. 1 2 CHAPTER 1. MAGNETIC MATERIALS â€¢ The important quantity for many purposes is the energy density of the magnet. 1.3 Magnetic properties - reminder â€¢ There are two ?elds to consider: â€“ The magnetic ?eld H which is generated by currents according to Ampèreâ€™s law. H is measured in A m -1 (Oersteds in old units) â€“ The magnetic induction, or magnetic ?ux density,B, which gives the energy of a dipole in a ?eld, E =-m.B and the torque experienced by a dipole momentm asG=m×B. B is measured in Wb m -2 or T (Gauss in old units). 1.3. MAGNETIC PROPERTIES - REMINDER 3 â€¢ In free space,B = µ 0 H. â€¢ In a material B = µ 0 (H+M) = µ 0 µ r H where µ r is the relative permeability, ? is the magnetic susceptibility, which is a dimensionless quantity. â€¢ Note, though, that ? is sometimes tabulated as the molar susceptibility ? m = V m ?, whereV m isthevolumeoccupiedbyonemole, orasthemass susceptibility ? g = ? ? , where ? is the density. â€¢ M, the magnetisation, is the dipole moment per unit volume. M= ?H. â€¢ In general, µ r (and hence ?) will depend on position and will be tensors (so thatB is not necessarily parallel toH). â€¢ Even worse, some materials are non-linear, so that µ r and ? are ?eld- dependent. â€¢ The e?ects are highly exaggerated in these diagrams. Page 4 Chapter 1 Magnetic Materials 1.1 Preliminaries 1.1.1 Required Knowledge â€¢ Magnetism â€¢ Electron spin â€¢ Atom â€¢ Angular momentum (quantum) â€¢ Statistical mechanics 1.1.2 Reading â€¢ Hook and Hall 7.1-7.3, 8.1-8.7 1.2 Introduction â€¢ Magnettechnologyhasmadeenormousadvancesinrecentyearsâ€“without the reductions in size that have come with these advances many modern devices would be impracticable. 1 2 CHAPTER 1. MAGNETIC MATERIALS â€¢ The important quantity for many purposes is the energy density of the magnet. 1.3 Magnetic properties - reminder â€¢ There are two ?elds to consider: â€“ The magnetic ?eld H which is generated by currents according to Ampèreâ€™s law. H is measured in A m -1 (Oersteds in old units) â€“ The magnetic induction, or magnetic ?ux density,B, which gives the energy of a dipole in a ?eld, E =-m.B and the torque experienced by a dipole momentm asG=m×B. B is measured in Wb m -2 or T (Gauss in old units). 1.3. MAGNETIC PROPERTIES - REMINDER 3 â€¢ In free space,B = µ 0 H. â€¢ In a material B = µ 0 (H+M) = µ 0 µ r H where µ r is the relative permeability, ? is the magnetic susceptibility, which is a dimensionless quantity. â€¢ Note, though, that ? is sometimes tabulated as the molar susceptibility ? m = V m ?, whereV m isthevolumeoccupiedbyonemole, orasthemass susceptibility ? g = ? ? , where ? is the density. â€¢ M, the magnetisation, is the dipole moment per unit volume. M= ?H. â€¢ In general, µ r (and hence ?) will depend on position and will be tensors (so thatB is not necessarily parallel toH). â€¢ Even worse, some materials are non-linear, so that µ r and ? are ?eld- dependent. â€¢ The e?ects are highly exaggerated in these diagrams. 4 CHAPTER 1. MAGNETIC MATERIALS 1.4 Measuring magnetic properties 1.4.1 Force method â€¢ Uses energy of induced dipole E =- 1 2 mB =- 1 2 µ 0 ?VH 2 , so in an inhomogeneous ?eld F =- dE dx = 1 2 µ 0 V? dH 2 dx = µ 0 V?H dH dx . â€¢ Practically: â€“ set up large uniformH; â€“ superpose linear gradient with additional coils â€“ vary second ?eld sinusoidally and use lock-in ampli?er to measure varying force 1.4.2 Vibrating Sample magnetometer â€¢ oscillate sample up and down â€¢ measure emf induced in coils A and B Page 5 Chapter 1 Magnetic Materials 1.1 Preliminaries 1.1.1 Required Knowledge â€¢ Magnetism â€¢ Electron spin â€¢ Atom â€¢ Angular momentum (quantum) â€¢ Statistical mechanics 1.1.2 Reading â€¢ Hook and Hall 7.1-7.3, 8.1-8.7 1.2 Introduction â€¢ Magnettechnologyhasmadeenormousadvancesinrecentyearsâ€“without the reductions in size that have come with these advances many modern devices would be impracticable. 1 2 CHAPTER 1. MAGNETIC MATERIALS â€¢ The important quantity for many purposes is the energy density of the magnet. 1.3 Magnetic properties - reminder â€¢ There are two ?elds to consider: â€“ The magnetic ?eld H which is generated by currents according to Ampèreâ€™s law. H is measured in A m -1 (Oersteds in old units) â€“ The magnetic induction, or magnetic ?ux density,B, which gives the energy of a dipole in a ?eld, E =-m.B and the torque experienced by a dipole momentm asG=m×B. B is measured in Wb m -2 or T (Gauss in old units). 1.3. MAGNETIC PROPERTIES - REMINDER 3 â€¢ In free space,B = µ 0 H. â€¢ In a material B = µ 0 (H+M) = µ 0 µ r H where µ r is the relative permeability, ? is the magnetic susceptibility, which is a dimensionless quantity. â€¢ Note, though, that ? is sometimes tabulated as the molar susceptibility ? m = V m ?, whereV m isthevolumeoccupiedbyonemole, orasthemass susceptibility ? g = ? ? , where ? is the density. â€¢ M, the magnetisation, is the dipole moment per unit volume. M= ?H. â€¢ In general, µ r (and hence ?) will depend on position and will be tensors (so thatB is not necessarily parallel toH). â€¢ Even worse, some materials are non-linear, so that µ r and ? are ?eld- dependent. â€¢ The e?ects are highly exaggerated in these diagrams. 4 CHAPTER 1. MAGNETIC MATERIALS 1.4 Measuring magnetic properties 1.4.1 Force method â€¢ Uses energy of induced dipole E =- 1 2 mB =- 1 2 µ 0 ?VH 2 , so in an inhomogeneous ?eld F =- dE dx = 1 2 µ 0 V? dH 2 dx = µ 0 V?H dH dx . â€¢ Practically: â€“ set up large uniformH; â€“ superpose linear gradient with additional coils â€“ vary second ?eld sinusoidally and use lock-in ampli?er to measure varying force 1.4.2 Vibrating Sample magnetometer â€¢ oscillate sample up and down â€¢ measure emf induced in coils A and B 1.5. EXPERIMENTAL DATA 5 â€¢ compare with emf in C and D from known magnetic moment â€¢ hence measured sample magnetic moment 1.5 Experimental data â€¢ In the ?rst 60 elements in the periodic table, the majority have negative susceptibility â€“ they are diamagnetic. 1.6 Diamagnetism â€¢ Classically, we have Lenzâ€™s law, which states that the action of a magnetic ?eld on the orbital motion of an electron causes a back-emf which opposes the magnetic ?eld which causes it. â€¢ Frankly, this is an unsatisfactory explanation, but we cannot do better until we have studied the inclusion of magnetic ?elds into quantum me- chanics using magnetic vector potentials. â€¢ Imagine an electron in an atom as a charge e moving clockwise in the x-y plane in a circle of radius a, area A, with angular velocity ?. â€¢ This is equivalent to a current I =charge/time= e?/(2p), so there is a magnetic moment µ= IA= e?a 2 /2. â€¢ The electron is kept in this orbit by a central force F = m e ? 2 a.Read More

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